Theorem ofcc 34212

Description: Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Hypotheses

Ref	Expression
ofcc.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofcc.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)
ofcc.3	⊢ (𝜑 → 𝐶 ∈ 𝑋)

Assertion

Ref	Expression
ofcc	⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))

Proof of Theorem ofcc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofcc.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2		fnconstg 6720	. . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
4		ofcc.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		ofcc.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
6		fvconst2g 7146	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
7	1, 6	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
8	3, 4, 5, 7	ofcfval 34204	. 2 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
9		fconstmpt 5684	. 2 ⊢ (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))
10	8, 9	eqtr4di 2787	1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {csn 4578   ↦ cmpt 5177   × cxp 5620   Fn wfn 6485  ‘cfv 6490  (class class class)co 7356   ∘_f/c cofc 34201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-ofc 34202

This theorem is referenced by: (None)